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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A214306 Triangle with entry a(n,m) giving the total number of bracelets of n beads (D_n symmetry) with n colors available for each bead, but only m distinct colors present, with m from {1, 2, ..., n} and n >= 1.

Original entry on oeis.org

1, 2, 1, 3, 6, 1, 4, 24, 24, 3, 5, 60, 180, 120, 12, 6, 165, 1120, 2040, 900, 60, 7, 336, 5145, 21420, 25200, 7560, 360, 8, 784, 23016, 183330, 442680, 335160, 70560, 2520, 9, 1584, 91056, 1320480, 5846400, 8890560, 4656960, 725760, 20160, 10
Offset: 1

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Author

Wolfdieter Lang, Jul 20 2012

Keywords

Comments

This triangle is obtained from the array A213941 by summing in row n, for n >= 1, all entries related to partitions of n with the same number of parts m.
a(n,m) is the total number of necklaces of n beads (dihedral D_n symmetry) corresponding to all the color multinomials obtained from all p(n,m) = A008284(n,m) partitions of n with m parts, written in nonincreasing form, by 'exponentiation'. Therefore only m from the available n colors are present, and a(n,m) gives the number of bracelets with n beads with only m of the n available colors present, for m from 1,2,...,n, and n >= 1. All of the possible color assignments are counted.
See the comments on A212359 for the Abramowitz-Stegun (A-St) order of partitions, and the 'exponentiation' to obtain multisets, used to encode color multinomials, from partitions. See a link in A213938 for representative multisets for given signature used to define color multinomials.
The row sums of this triangle coincide with the ones of array A213941, and they are given by A081721.

Examples

			n\m 1    2     3       4       5       8       7      8     9
1   1
2   2    1
3   3    6     1
4   4   24    24       3
5   5   60   180     120      12
6   6  165  1120    2040     900     60
7   7  336  5145   21420   25200   7560      360
8   8  784 23016  183330  442680  33516    70560   2520
9   9 1584 91056 1320480 5846400 8890560 4656960 725760 20160
...
Row n=10:  10, 3420, 357480, 8691480, 64420272, 172609920, 177811200, 68040000, 8164800, 181440;
Row n=11:  11, 6820, 1327095, 52727400, 622175400, 2714009760, 4837417200, 3592512000, 1047816000, 99792000, 1814400.
a(2,2) = 1 from the color monomial c[1]^1*c[2]^1 = c[1]*c[2] (from the m=2 partition [1,1] of n=2). The bracelet in question is cyclic(12) (we use j for color c[j] in these examples). The same holds for the necklace case.
a(5,3) = 60 + 120 = 180, from A213941(5,4) + A213941(5,5), because k(5,3,1) = A214314(5,3) = 4 and p(5,3)=2.
a(3,1) = 3 from the color monomials c[1]^3, c[2]^3 and c[3]^3. The three bracelets are cyclic(111), cyclic(222) and cyclic(333). The same holds for the necklace case.
In general a(n,1)=n from the partition [n] providing the color signature (exponent), and the n color choices.
a(3,2) = 6 from the color signature c[.]^2 c[.]^1, (from the m=2 partition [2,1] of n=3), and there are 6 choices for the color indices. The 6 bracelets are cyclic(112), cyclic(113), cyclic(221), cyclic(223), cyclic(331) and cyclic(332). The same holds for the necklace case.
a(3,3) = 1. The color multinomial is c[1]*c[2]*c[3] (from the m=3 partition [1,1,1]). All three available colors are used. There is only one bracelet: cyclic(1,2,3). The necklace cyclic(1,3,2) becomes equivalent under D_3 operation.
a(4,2) = 24 from two color signatures c[.]^3 c[.] and c[.]^2 c[.]^2 (from the two m=2 partitions of n=4: [3,1] and [2,2]). The first one produces 4*3=12 bracelets, namely 1112, 1113, 1114, 2221, 2223, 2224, 3331, 3332, 3334, 4441, 4442 and 4443, all taken cyclically. The second color signature leads to another 6*2=12 bracelets: 1122, 1133, 1144, 2233, 2244, 3344, 1212, 1313, 1414, 2323, 2424 and 3434, all taken cyclically. Together they provide the 24 bracelets counted by a(4,2). The same holds for the necklace case.
a(4,3) = 24 from the color signature c[.]^2 c[.]c[.]. There are 4*3 =12 color choices each with two bracelets: 1123, 1213, 1124, 1214, 1134, 1314, 2213, 2123, 2214, 2124, 2234, 2324, 3312, 3132, 3314, 3134, 3324, 3234, 4412, 4142, 4413, 4143, 4423 and 4243, each taken cyclically.

Links

Crossrefs

Cf. A081721, A212360 (necklaces), A213941, A273891.

Programs

  • Mathematica
    (* t = A081720 *) t[n_, k_] := (For[t1 = 0; d = 1, d <= n, d++, If[Mod[n, d] == 0, t1 = t1 + EulerPhi[d]*k^(n/d)]]; If[EvenQ[n], (t1 + (n/2)*(1 + k)*k^(n/2))/(2*n), (t1 + n*k^((n + 1)/2))/(2*n)]);
T[n_, k_] := Binomial[n, k]*Sum[(-1)^i * Binomial[k, i]*t[n, k - i], {i, 0, k - 1}];
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Oct 08 2017, after Andrew Howroyd *)

Formula

a(n,m) = Sum_{j=1..p(n,m)} A213941(n, k(n,m,1)+j-1), with k(n,m,1) = A214314(n,m) the position where in the list of partitions of n in A-St order the first with m parts appears, and p(n,m) is the number of partitions of n with m parts shown in the array A008284. E.g., n=5, m=3: k(5,3,1) = A214314(5,3) = 4, p(5,3) = 2.
a(n,m) = binomial(n,m) * A273891(n,m). - Andrew Howroyd, Mar 25 2017

A214311 a(n) is the number of representative five-color bracelets (necklaces with turning over allowed) with n beads, for n >= 5.

Original entry on oeis.org

12, 30, 150, 633, 3260, 16212, 66810, 298495, 1410402, 6403842, 31103899, 135342046, 633228696, 2936824916, 13676037486, 65355191817, 298065986582, 1398226666434, 6585151203697, 30958838054304, 148994847644780
Offset: 5

Views

Author

Wolfdieter Lang, Aug 08 2012

Keywords

Comments

This is the fifth column (m=5) of triangle A213940.
The relevant p(n,5)= A008284(n,5) representative color multinomials have exponents (signatures) from the five-part partitions of n, written with nonincreasing parts. E.g., n=7: [3,1,1,1,1] and [2,2,1,1,1] (p(7,5)=2). The corresponding representative bracelets have the five-color multinomials c[1]^3*c[2]*c[3]*c[4]*c[5] and c[1]^2*c[2]^2*c[3]*c[4]*c[5].
Number of n-length bracelets w over a 5-ary alphabet {a1,a2,...,a5} such that #(w,a1) >= #(w,a2) >= ... >= #(w,a5) >= 1, where #(w,x) counts the letters x in word w (bracelet analog of A226884). The number of 5 color bracelets up to permutations of colors is given by A056360. - Andrew Howroyd, Sep 26 2017

Examples

			a(5) = A213940(5,5) = A213939(5,7) = 12 from the representative bracelets (with colors j for c[j], j=1,...,5) 12345, 12354, 12435, 12453, 12534, 12543, 13245, 13254, 13425, 13524, 14235 and 14325, all taken cyclically.

Links

Crossrefs

Cf. A213939, A213940, A214309 (m=4 case), A214313 (m=5, all bracelets).
Cf. A056360, A226884.

Formula

a(n) = A213940(n,5), n >= 5.
a(n) = sum(A213939(n,k),k= b(n,5)..b(n,6)-1), n>=6, with b(n,m) = A214314(n,m) the position where the first m part partition of n appears in the Abramowitz-Stegun ordering of partitions (see A036036 for the reference and a historical comment). a(5) = A213939(5,b(5,5)) = A213939(5,7) = 12.

A214312 a(n) is the number of all four-color bracelets (necklaces with turning over allowed) with n beads and the four colors are from a repertoire of n distinct colors, for n >= 4.

Original entry on oeis.org

3, 120, 2040, 21420, 183330, 1320480, 8691480, 52727400, 303958710, 1674472800, 8928735816, 46280581620, 234611247780, 1166708558400, 5710351190400, 27565250985360, 131495088522060, 620771489730000, 2903870526350640, 13473567673441260, 62061657617625204, 283995655732351200
Offset: 4

Views

Author

Wolfdieter Lang, Jul 31 2012

Keywords

Comments

This is the fourth column (m=4) of triangle A214306.
Each 4 part partition of n, with the parts written in nonincreasing order, defines a color signature. For a given color signature, say [p[1], p[2], p[3], p[4]], with p[1] >= p[2] >= p[3] >= p[4] >= 1, there are A213941(n,k)= A035206(n,k)*A213939(n,k) bracelets if this signature corresponds (with the order of the parts reversed) to the k-th partition of n in Abramowitz-Stegun (A-St) order. See A213941 for more details. Here all p(n,4)= A008284(n,4) partitions of n with 4 parts are considered. The color repertoire for a bracelet with n beads is [c[1], ..., c[n]].
Compare this with A032275 where also bracelets with less than four colors are included, and the color repertoire is only [c[1], c[2], c[3], c[4]] for all n.

Examples

			a(5) = A213941(5,6) = 120 from the bracelet (with colors j for c[j], j=1, 2, ..., 5) 11234, 11243, 11324, 12134, 13124 and 14123, all six taken cyclically, each representing a class of order A035206(5,6) = 20 (if all 5 colors are used). For example, cyclic(11342) becomes equivalent to cyclic(11243) by turning over or reflection. The multiplicity 20 depends only on the color signature.

Links

Crossrefs

Cf. A213941, A214306, A214309 (m=4, representative bracelets), A214313 (m=5).

Programs

  • Mathematica
    t[n_, k_] := (For[t1 = 0; d = 1, d <= n, d++, If[Mod[n, d] == 0, t1 = t1 + EulerPhi[d]*k^(n/d)]]; If[EvenQ[n], (t1 + (n/2)*(1 + k)*k^(n/2))/(2*n), (t1 + n*k^((n + 1)/2))/(2*n)]);
a56344[n_, k_] := Sum[(-1)^i*Binomial[k, i]*t[n, k - i], {i, 0, k - 1}];
a[n_] := Binomial[n, 4]*a56344[n, 4];
Table[a[n], {n, 4, 25}] (* Jean-François Alcover, Jul 02 2018, after Andrew Howroyd *)

Formula

a(n) = A214306(n,4), n >= 4.
a(n) = sum(A213941(n,k),k = A214314(n,4) .. (A214314(n,4) - 1 + A008284(n,4))), n >= 4.
a(n) = binomial(n,4) * A056344(n). - Andrew Howroyd, Mar 25 2017

A214310 a(n) is the number of all three-color bracelets (necklaces with turning over allowed) with n beads and the three colors are from a repertoire of n distinct colors, for n >= 3.

Original entry on oeis.org

1, 24, 180, 1120, 5145, 23016, 91056, 357480, 1327095, 4893680, 17525508, 62254920, 217457695, 753332160, 2581110000, 8779264032, 29624681763, 99350001360, 331159123260, 1098168382080, 3624003213369, 11908069219816, 38972450763000, 127087400895000
Offset: 3

Views

Author

Wolfdieter Lang, Jul 31 2012

Keywords

Comments

This is the third column (m=3) of triangle A214306.
Each 3 part partition of n, with the parts written in nonincreasing order, defines a color signature. For a given color signature, say [p[1], p[2], p[3]], with p[1] >= p[2] >= p[3] >= 1, there are A213941(n,k)= A035206(n,k)* A213939(n,k) bracelets if this signature corresponds (with the order of the parts reversed) to the k-th partition of n in Abramowitz-Stegun (A-St) order. See A213941 for more details. Here all p(n,3)= A008284(n,3) partitions of n with 3 parts are considered. The color repertoire for a bracelet with n beads is [c[1], ..., c[n]].
Compare this with A027671 where also single color bracelets are included, and the color repertoire is only [c[1], c[2], c[3]] for all n.

Examples

			a(5) = A213941(5,4) + A213941(5,5) = 60 + 120 = 180 from the bracelet (with colors j for c[j], j=1, 2, ..., 5) 11123 and 11213, both taken cyclically, each representing a class of order A035206(5,4)= 30 (if all 5 colors are used), and 11223, 11232, 12123 and 12213, all taken cyclically, each representing a class of order A035206(5,5)= 30. For example, cyclic(11322) becomes equivalent to cyclic(11223) by turning over or reflection. The multiplicity A035206 depends only on the color signature.

Links

Crossrefs

Cf. A213941, A214306, A214307 (m=3, representative bracelets), A214312 (m=4).

Formula

a(n) = A214306(n,3), n >= 3.
a(n) = sum(A213941(n,k), k = A214314(n,3).. (A214314(n,3) - 1 + A008284(n,3))), n >= 3.
a(n) = binomial(n,3) * A056343(n). - Andrew Howroyd, Mar 25 2017

Extensions

a(26) from Andrew Howroyd, Mar 25 2017

A214313 a(n) is the number of all five-color bracelets (necklaces with turning over allowed) with n beads and the four colors are from a repertoire of n distinct colors, for n >= 5.

Original entry on oeis.org

12, 900, 25200, 442680, 5846400, 64420272, 622175400, 5466166200, 44611306740, 343916472900, 2531921456064, 17956666859040, 123458676825120, 827056125453600, 5419508203393200, 34847210197637424, 220424306985639540, 1374479672119161300, 8463477229726134000, 51536194734146965920, 310706598354410079360
Offset: 5

Views

Author

Wolfdieter Lang, Aug 08 2012

Keywords

Comments

This is the fifth column (m=5) of triangle A214306.
Each 5 part partition of n, with the parts written in nonincreasing order, defines a color signature. For a given color signature, say [p[1], p[2], ..., p[5]], with p[1] >= p[2] >= .. >= p[5] >= 1, there are A213941(n,k) = A035206(n,k)*A213939(n,k) bracelets if this signature corresponds (with the order of the parts reversed) to the k-th partition of n in Abramowitz-Stegun (A-St) order. See A213941 for more details. Here all p(n,5)= A008284(n,5) partitions of n with 5 parts are considered. The color repertoire for a bracelet with n beads is [c[1], ..., c[n]].
It appears that this sequence is divisible by 12, producing 1, 75, 2100, 36890, 487200, 5368356, 51847950, 455513850, ...
Compare this with A056345 where only 5 colors are used for all n >= 5.

Examples

			a(6) = A213941(6,10) = 900 from the bracelet with color signature [2,1,1,1,1] and color repertoire [c[j], j=1, 2, ..., 6]. There are A213939(6,10) = 30 bracelets with representative color multinomials c[1]^2 c[2] c[3] c[4] c[5]. If the colors c[j] are taken as j, e.g., 112345, 112354, 112435, 112453, 112534, 112543, 113245, 113254, 113425, (113452 is equivalent to 112543 by turning over), 113524, (113542 ==112453), 114235, ..., 121345, ... (all taken cyclically). Each of these 30 bracelets represents a class of A035206(6,10) = 30 bracelets when all six colors are used. Thus a(6) = 30*30 = 900 = 12*75.

Links

Crossrefs

Cf. A213941, A214306, A214311 (m=5, representative bracelets), A214312 (m=4).

Formula

a(n) = A214306(n,5), n >= 5.
a(n) = sum(A213941(n,k),k = A214314(n,5) .. (A214314(n,5) - 1 + A008284(n,5))), n >= 5.
a(n) = binomial(n,5) * A056345(n). - Andrew Howroyd, Mar 25 2017

A292223 a(n) is the number of representative six-color bracelets (necklaces with turning over allowed; D_6 symmetry) with n beads, for n >= 6.

Original entry on oeis.org

60, 180, 1050, 5040, 29244, 161340, 1046250, 4825800, 27790266, 145126548, 843333015, 4466836920, 26967624184, 137243187108, 789854179074, 4306147750200, 24711052977222, 134216193832908, 797987818325009, 4240082199867228
Offset: 6

Views

Author

Wolfdieter Lang, Sep 30 2017

Keywords

Comments

This is the sixth column (m = 6) of triangle A213940.
The relevant p(n,6)= A008284(n, 6) representative color multinomials have exponents (signatures) from the six-part partitions of n, written with nonincreasing parts. E.g., n = 8: [3,1,1,1,1,1] and [2,2,1,1,1,1] (p(8,6)=2). The corresponding representative bracelets have the six-color multinomials c[1]^3*c[2]*c[3]*c[4]*c[5]*c[6] and c[1]^2*c[2]^2*c[3]*c[4]*c[5]*c[6].
See A056361 for the numbers if also color permutations for D_6 inequivalent bracelets are allowed. (Andrew Howroyd induced me to look at these bracelets.)

Examples

			a(6) = A213940(6,6) = A213939(6, 11) = 60 from the representative bracelets (with colors j for c(j), j=1..6) permutations of (1, 2, 3, 4, 5, 6) modulo D_6 (dihedral group) symmetry, i.e., modulo cyclic or anti-cyclic operations. E.g., (1, 2, 3, 4, 6, 5) == (2, 3, 4, 6, 5, 1) == (6, 4, 3, 2, 1, 5) == ..., but (1, 2, 3, 4, 6, 5) is not equivalent to (1, 2, 3, 4, 5, 6). If color permutation is also allowed, then there is only one possibility (see A056361(6) = 1).

Crossrefs

Cf. A008284, A036036, A056361, A213940, A214311, A214314.

Formula

a(n) = A213940(n, 6), n >= 6.
a(n) = Sum_{k=b(n, 6)..b(n, 7)-1} A213939(n, k), for n >= 7, with b(n, m) = A214314(n, m) the position where the first m-part partition of n appears in the Abramowitz-Stegun ordering of partitions (see A036036 for the reference and a historical comment), and a(6) = A213939(6, b(6,6)) = A213939(6, 11) = 60.
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